Sport Science Looks at the Vertical Leap

Ok, well here is the episode. It looks at the vertical jumping height of Dwight Howard. If you don’t want to watch it, I will summarize.

Here is the play-by-play:

Shaq can jump super high.

Dwight Howard can jump super high.

Let’s see if Howard can break Shaq’s record for vertical “reach”.

Apparently he can. His vertical reach (how high his hand gets on the backboard) gets up to 12 feet 6 inches. This is a leap of 39 inches.

Really, Sport Science could have made a nice episode here. However, it is possible they took it too far. After showing Howard and his awesome jump, they try to add some science. I guess they are contractually obliged to have some numbers in there (and pointless sensors) or else they would be called “ESPN Sport” instead of “ESPN Sport Science”.

Here is the summary of the science part:

Let’s put these pressure sensors in Howard’s shoes so that we can measure force and make a cool pressure-position graph.

The force on his feet is reported as 1,210 pounds of “push”. I am not sure if this is the average force or the maximum. Clearly that type of information isn’t important.

Howard produces 1,506 Watts of power in the jump. No idea how they came up with this number – but I am going to check it.

If some smaller guy had the same power as Howard, he would have a vertical leap of 61 inches. I think I should check this too.

Clearly, some things need to be checked.

Jumping Physics

How much force would you need to push on the ground to jump a certain height? Let’s make an estimate. Here is a diagram of a jumping player.

I assume vertical leap is the change in height of the center of mass of the jumper from standing to the highest point in the jump. However, you can’t jump like this. So, the middle diagram shows the jumper right before starting the jumper. In this case, the jumper is both moving and bending the legs. During the jump, the jumper pushes on the floor and moves the center of mass up a distance s also the jumper “turns” so that this forward speed v1 is used to help go even higher.

Now for the physics. In a case like this where both force and distances are important, the Work-Energy principle is the one to use. I can write work and energy as:

Here s is the distance through which the force acts on the object and ? is the angle between the force and the direction the object moves. I will look at this work as the jumper goes from the moving squatting position to the highest point (and not moving – or moving very little). Now, there is one little “cheat” I am going to use. I want to include the force the jumper pushes on the ground. However, this force doesn’t have a displacement (the floor doesn’t move when pushing on the feet). In this case, the force of the floor would not do any work. So here is the trick. I will pretend like the force the floor pushes on the person moves as the center of mass moves. If I do this, I can calculate the work done by the floor on the person (even though it is really the person doing work on the person by using muscles).

Since there are two forces on the jumper (gravity and the floor), I will have:

Now what? Let me check their value of the jumping force by using their force and determining the distance s that the jumper would have to go down in his pre-jump. I will need a couple of other things to estimate (or look up).

Starting speed of about 3 m/s.

Mass of jumper. For Dwight Howard, this would be 120 kg.

h is 0.99 meters.

Oh, and the force is 1,210 pounds or 5382 Newtons.

Solving for the jump distance s:

Using the values above, I get a value of 14.8 cm for the amount of “squatting”. That seems a bit low – but not crazy. This could be the average force. Well done Sport Science.

Power and Jumping

Next, for the power. Power is defined as:

Personally, I wouldn’t think that power is the best way to characterize this jump. Why? Because no one recorded the time that they jumper jumps. Oh well, I didn’t write the show.

But, if it is power they want – it is power they will get. I can get power. How? I just need the time. If I assume a constant acceleration during the jump, I can use the average velocity (vertical velocity) to find the time. Step one: find the vertical velocity right when leaving the ground.

Again, I can use the same ideas as before. However, instead of starting with the beginning of the jump and ending at the highest point, I will start right AFTER the jump and end at the highest point. So, there is no force from the floor since this is after that. Here is the work-energy expression for this motion.

I know you are worried about the starting horizontal speed, but remember this is the vertical velocity and this is what matters since I am looking at the change in vertical position (s). This would give time to jump of:

Now that I have the time, I just need the energy. How much work do you need to get the jumper up to some height? I just need to go back to the work-energy (again):

Putting this in with the time, I get a power of:

Using the values from before, this gives an average power of 12.4 kWatts. That is some serious power – but just for 0.06 seconds. However, this is not the value sport science gives. Hmmm. This is like 10 times the value reported in the video. Why are they different? My first guess is that the value I obtained for s is too small. However, if I double this the power is only about half as much.

My next thought: how did Sport Science get the 1500 Watts value? Maybe they calculated the power using the work done by gravity for the rising part divided by the time he was in the air? Using this kinematic equation, I get can the time for half of the jumper’s flight:

This would be a total flight time of 0.89 seconds. The energy needed to get that high (neglecting the pre-run part) would just be mgh or about 1164 Joules. Dividing energy by this time gives a power of 1295 Watts. It is scary that this is pretty close to the value reported in the video. Please tell me this is not what they did. Please. Please. If you just use slightly different numbers (like from silly rounding) you can get the exact same value Sport Science lists. Scary.

Using the time of the flight (time the jumper is in the air) is totally wrong. Why? This is the time where the person is just doing nothing except falling. The jumper isn’t using muscles at all. This has nothing to do with the jumper’s muscles. I could throw a 120 kg bag of sand the same height, but the power would all come from the throw, not the flight.

A Jumping Comparison

In true Sport Science fashion, you have to make some comparison at the end of the show (again, it is in the contract). For this episode they claim that if Nate Robinson (a smaller guy) had the same power, he would jump 61 inches (1.55 meters). I hate to do this, but let me just use the wrong method I suspect they used above. If I get the same answer Sport Science states, that pretty much confirms that they did it that way.

According to wikipedia, Robinson is 1.75 meters tall with a mass of 82 kg. Ok – to make this work, I will use the same power AND the same time as Dwight Howard (although with shorter legs I suspect the time to jump would be shorter also). Quick note – subscripts of H will be used for Howard and R for Robinson.

I get 57 inches compared to the value from Sport Science of 61 inches. I suspect they used the same idea, but with a slightly different calculation. Actually, I see the problem. Here it is:

Sport Science was using a weight of 280 pounds (127 kg). I had been using the wikipedia listed mass of 120 kg. Using this mass, I get a wrong height of 60.4 inches. I am pretty sure this is what they did. BAD SPORT SCIENCE.

Well, how would you calculate the height with the same power? I would say that the jumping height (and thus time) are a bit shorter and go back to the power-jumping stuff above. Of course, they are still using the wrong power.

In the end, this is another case of Sport Science just making up some stuff and calling it science. Why do they do that?

If I ran the Sport Science

I would make a few changes, that’s just what I would do – said a young Gerald McGrew (that is from Dr. Seuss). Really, there is some nice potential here. People like sports and there are some interesting possible questions. It could work. Here are some suggested changes for this episode:

Never EVER use some sensor just because it looks cool. Don’t make this animated skeleton of the sports person for no reason other than looking cool. In this case, it seems clear that the pressure sensor was just for looks.

Don’t make crazy comparisons between things that can’t be compared.

You could talk about why taller jumpers can jump higher. Not only do they start higher, they have a longer displacement during the jump motion.

A graph showing some scaling of jump height vs. jumper height would be cool.

If you want some science, clearly show the force (not the max force) that the jumper exerts on the floor along with the displacement of the jumper. Talk about how the larger the force or displacement, the greater the change in energy.

Let me end with a note to Sport Science.

Dear Sport Science,

You have a show. Clearly, it involves sports but it rarely involves science. Why? Why would you do that?

In the interest of science and for the world, I will gladly help you. Next time you want to do something, give me a buzz. I will even do your calculations for you. Just please stop the insanity. Please. You aren’t helping any.